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July 20th, 2016, 07:30 AM   #1
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Number Systems and Basis of Real Analysis

Decimals, Points on a Unit Line, and the Natural Numbers

Start with 1 and put n-1 zeroes in front of it and a decimal point, .00001 for example. This defines a division of the unit line into 10^5 increments. The increment points on the line can then be labelled and counted off sequentially:
0
.00001
.00002
.00003
.......
.99999
1
As n approaches infinity, the points on the line approach all the real numbers and the numbering of the points on the line approaches the set of all the natural numbers.

The Natural Numbers, Points on a Unit Line, and the Continuum

Every natural number q corresponds to a division of a unit line into q segments. Label the increment points p/q, p=1...q. This establishes a direct connection between the natural numbers, points on a unit line, and the continuum as q approaches infinity. The decimal system above is the special case of q=10^n. q could also be 2^n, 3^n, 4^n,....

A point of 0 width doesn't exist any more than infinity exists, other than as a concept or a notation.

Infinity means countable infinity, of course. A unit line is any line you call 1, the meter for example.

Last edited by skipjack; July 20th, 2016 at 11:05 AM.
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July 20th, 2016, 09:02 AM   #2
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Moved from thread about books and source material Zylo...
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Last edited by skipjack; July 20th, 2016 at 11:09 AM.
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July 20th, 2016, 10:37 AM   #3
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Quote:
Originally Posted by zylo View Post
As n approaches infinity, the points on the line approach all the real numbers and the numbering of the points on the line approaches the set of all the natural numbers.
This is a blind guess on your part. You offer it without any supporting argument and it is wrong on three counts. $n$ cannot approach infinity, the points do not approach all the real numbers and the numbering does not approach the natural numbers.

Yet again you have demonstrated your ignorance of the difference between potential infinity and actual infinities.

You also demonstrate your ignorance of logic, since your ravings do not indicate any flaws in existing proofs that contradict your result.

Quote:
Originally Posted by zylo View Post
A point of 0 width doesn't exist any more than infinity exists, other than as a concept
Yes, points exist as concepts. That is exactly what they are. Similarly for infinities. Both exist in appropriate contexts. All of mathematics is conceptual. If you want more physical constructs, you must study physics.
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Last edited by skipjack; July 20th, 2016 at 11:04 AM.
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July 20th, 2016, 10:14 PM   #4
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Well, Archie you were right. He is dealing with limits, at least in this post.

I am getting amused by the looseness of his arguments.

As we increase the number of cuts on the line, we obviously get closer to a greater number of reals. Start with 0.5. Now add cuts at 0.25 and 0.75. Obviously all real numbers less that 0.375 are closer to 0.25 than to 0.5. The distance has been reduced from < 0.5 to < 0.25.

Now conceive of a denumerably infinite number of such cuts, dividing the line into a denumerably infinite number of open intervals. (I think you are saying such a conception is invalid, but let's accept it.)

When he says "This establishes ..." nothing has been established except a denumerably infinite number of cuts. Effectively he is assuming one real number in each cut. If that were so, he would have shown the required 1-to-1 correspondence.. But he has no argument whatsoever on why there is only one real number in each cut. In fact, he does not even explicitly state this crucial step. He implicitly assumes that which is to be proven.
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July 20th, 2016, 11:55 PM   #5
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Originally Posted by JeffM1 View Post
Effectively he is assuming one real number in each cut. If that were so, he would have shown the required 1-to-1 correspondence.. But he has no argument whatsoever on why there is only one real number in each cut.
It's worth pointing out that there are an infinite number of rational numbers between each pair of cuts, so Zylo's suggestion that we could in any way get to having one real number per cut is utter nonsense.

In terms of Zylo's beloved limits, the quantity of numbers between each pair of cuts is constant for each $n$, so the limit as $n$ grows without bound is that same infinite quantity. This is just as $\lim \limits_{n \to \infty} a = a$ for any constant $a$. (Of course, in this case we kind of have $a=\mathfrak c$ or $a = \aleph_0$ depending on whether we are counting reals or naturals and it's highly debatable how much sense that makes, but that's what happens when you take nonsense as the starting point for a discussion).

Last edited by skipjack; July 21st, 2016 at 03:19 AM.
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July 21st, 2016, 12:15 AM   #6
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Originally Posted by JeffM1 View Post
Now conceive of a denumerably infinite number of such cuts, dividing the line into a denumerably infinite number of open intervals. (I think you are saying such a conception is invalid, but let's accept it.)
Some research suggests that it is possible to create a partition of an interval into uncountably many uncountable subsets, but I don't think the way that Zylo describes could ever do that.

However, it doesn't change that his method doesn't get us to the set of points coinciding with the set of reals.

Last edited by skipjack; July 21st, 2016 at 03:20 AM.
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July 21st, 2016, 02:27 AM   #7
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Kudos to SkipJack for moving the thread!
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July 21st, 2016, 03:04 AM   #8
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What's also amusing is everyone's reference to Zylo in third person. As if he's a bot or something... Oh wait..
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July 21st, 2016, 06:17 AM   #9
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There's no point in directing comments at him, because he just ignores everything that disagrees with his fantasy.
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